We give a detailed discussion about existence and uniqueness of Lu's momentum map. More precisely, we introduce the infinitesimal momentum map, and we study its properties. This allows us to describe the theory of reconstruction of the momentum map from the infinitesimal one. We provide the conditions for the uniquenes…
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Constructibility of momentum maps and linear variation for singular symplectic reduced spacesmath.SG
The paper extends a theorem about momentum maps to singular symplectic spaces.
problem Extending a theorem about momentum maps to singular symplectic spaces.
method Using integral affine stratification and equivariant locally trivial fibrations, the paper extends the linear variation theorem to singular values of the momentum map.
result Cohomology classes of symplectic forms on reduced spaces vary linearly within strata.
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and -algebras.
result Extension of conserved currents to spacetime vector fields.
This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to , the bundle of vertically adapted linear frames over the bundle of field configurations . Specifically, the generalized field momentum obs…
In this thesis we study the classical and quantum momentum maps and the theory of reduction. We focus on the notion of momentum map in Poisson geometry and we discuss the classification of the momentum map in this framework. Furthermore, we describe the so-called Poisson Reduction, a technique that allows us to reduce …
Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.
Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
This paper simplifies complex nonholonomic systems using momentum map reduction.
problem Reducing complex nonholonomic systems with symmetries.
method Using nonholonomic momentum bundle map and gauge transformation.
result Reduced manifolds are Chaplygin-type leaves with an almost symplectic form.
Generalizes momentum map to Courant algebroid for constrained mechanics.
problem Generalizing momentum map to new geometric structures.
method Generalized momentum section on Lie algebroid to Courant algebroid, constructed cohomological formulations.
result Identified momentum section in constrained Hamiltonian mechanics with Courant algebroid symmetry.
The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more g…
The paper introduces a new form on Lie algebroids over multisymplectic manifolds.
problem Higher generalizations of Poisson structures and momentum maps.
method Introducing a compatible E-n-form on Lie algebroids.
result The introduced form satisfies a compatibility condition with Lie algebroid and multisymplectic structures.
A symplectic slice theoremmath.SG
We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a cru…
In this article a homotopy co-momentum map (à la Callies-Frégier-Rogers-Zambon) trangressing to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coa…
Introduces comomentum sections and proves they are Poisson maps.
problem Generalizing Poisson maps to Hamiltonian Lie algebroids.
method Introduces comomentum sections and proves they are Lie algebroid morphisms and Poisson maps.
result Comomentum sections are Poisson maps between proper Poisson manifolds.
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three appr…
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentu…
Study norm-squared of momentum map in infinite dimensions with applications to symplectic geometry.
problem Understanding the norm-squared of the momentum map in infinite-dimensional settings.
method Calculation of Hessian, decomposition of stabilizer, application to symplectic and complex structures.
result Positive semi-definiteness of Hessian along complexified orbit and new central extensions of symplectomorphism group.
Let a torus T act effectively on a compact connected cooriented contact manifold, and let Psi be the natural momentum map on the symplectization. We prove that, if dim T > 2, the union of the origin with the image of Psi is a convex polyhedral cone, the non-zero level sets of Psi are connected (while the zero level set…
Poisson Reductionmath.DG
In this paper we develope a theory of reduction for classical systems with Poisson Lie groups symmetries using the notion of momentum map introduced by Lu. The local description of Poisson manifolds and Poisson Lie groups and the properties of Lu's momentum map allow us to define a Poisson reduced space.
The optimal momentum mapmath.SG
The presence of symmetries in a Hamiltonian system usually implies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden--Weinstein reduction procedure takes advantage of this and associates to the…
In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, t…
Triangle groups in the complex hyperbolic plane and special fibers of the momentum map in PU(2,1)math.DG
Study character varieties for 3-punctured sphere group representations in PU(2,1).
problem Characterize representations with special elliptic elements.
method Analyze relative character varieties and momentum maps.
result Representations form fibers of the momentum map.
Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.
problem Reduction of controlled Lagrangian systems with symmetry and momentum map.
method Using Legendre transformation and Euler-Lagrange vector field, the paper extends symmetric reduction theory.
result Established regular reduction theory for RCL systems with symmetry and momentum map.
This paper analyzes two Lie group momentum optimization algorithms and their convergence rates.
problem Optimizing functions on Lie groups using momentum-based dynamics.
method Investigates Lie Heavy-Ball and Lie NAG-SC algorithms, quantifying their convergence rates under smoothness and convexity assumptions.
result Lie NAG-SC accelerates optimization over the momentumless case, while Lie Heavy-Ball does not.
We present a reduction procedure for locally conformally symplectic (LCS) manifolds with an action of a Lie group preserving the conformal structure, with respect to any regular value of the momentum mapping. Under certain conditions, this reduction is compatible with the existence of a locally conformally Kähler struc…
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Ca…
Momentum ResNets improve ResNets' memory efficiency.
problem Memory inefficiency in deep residual neural networks (ResNets).
method Adding a momentum term to the forward rule of ResNets to make them invertible.
result Momentum ResNets can learn any linear mapping up to a multiplicative factor, improving memory efficiency.
The paper simplifies complex mechanical systems with external forces.
problem Analyzing symmetric discrete mechanical systems with external forces.
method Lagrangian reduction and reconstruction for principal bundles.
result Evolution of momentum maps and Poisson structures under different conditions.
Bayesian method learns optimal momentum for landmark matching.
problem Finding a diffeomorphism between two sets of landmarks.
method Ensemble Kalman filter for derivative-free Bayesian inverse method.
result Efficient algorithm for various target shapes.
Establishes inequality for multiple black holes, proving mass lower bound.
problem Proving mass-angular momentum inequality for multiple black holes.
method Novel flow of singular harmonic maps with hyperbolic plane target.
result Mass lower bound for ADM mass and angular momentum.
Paper develops momentum schemes with variance reduction for non-convex composition optimization.
problem Lack of convergence guarantee and efficient momentum design in existing algorithms.
method Develops various momentum schemes with SPIDER-based variance reduction.
result Achieves near-optimal sample complexity and linear convergence rate.
These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in computational anatomy.
We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…
Study the commutativity of reduction and symplectification in contact Hamiltonian systems.
problem Understanding the commutativity of reduction and symplectification in contact Hamiltonian systems.
method Introduce symplectic and contact geometry, perform reduction via momentum map, analyze symplectification process.
result Commutativity relations between reduction and symplectification in contact Hamiltonian systems.
Theory of symplectic reduction in infinite dimensions developed.
problem Challenges in symplectic reduction in infinite dimensions.
method Normal form of momentum map for infinite-dimensional equivariant maps.
result Theory of singular symplectic reduction in infinite dimensions.
The Area-Angular Momentum-Charge Inequality for Black Holes With Positive Cosmological Constantgr-qc
We establish the conjectured area-angular momentum-charge inequality for stable apparent horizons in the presence of a positive cosmological constant, and show that it is saturated precisely for extreme Kerr-Newman-de Sitter horizons. As with previous inequalities of this type, the proof is reduced to minimizing an `ar…
Paper extends Noether's Theorem to nonholonomic systems, proving conserved momentum.
problem Link between symmetries and first integrals broken in nonholonomic systems.
method Constructive method proving conserved momentum under certain conditions.
result Conserved momentum map in nonholonomic systems, extending Noether's Theorem.
We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the…
A new principle for optimizer selection improves training speed and performance.
problem Finding the best optimizer hyperparameters for faster training.
method Formulate optimizer selection as maximizing the expected drop rate in loss, treating gradients and updates as signals and an optimizer as a causal filter.
result Greedy optimizer selection yields stable and effective momentum rules.
We prove a criterion for stability of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. This generalizes a theorem of G.W. Patrick. The method of the proof is also useful in studying the bifurcation of relative equilibria.
We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and hamiltonian actions, to the realm of Dirac geometry. As an example, we show how hamiltonian quasi-Poisson manifolds fit into this framework by constructing an ``inversion'' procedur…
We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map.
I present a class of examples of \CR-submanifolds of manifolds endowed with different structures, obtained as level sets of momentum maps associated to specific Hamiltonian actions.
Classifies scalar-flat toric Kähler instantons in 4D.
problem Classifying scalar-flat toric Kähler 4-manifolds.
method Using Liouville theorem for degenerate-elliptic equations, classifies momentum functions and metrics.
result Fully classifies instantons with ALE-F-G-H asymptotic types.
Optimal reductionmath.SG
We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit red…
Study of harmonic maps with extreme Kerr-like singularities.
problem Analyzing harmonic maps with specific singularities.
method Asymptotic analysis of harmonic maps from 3D Euclidean space to hyperbolic plane.
result Existence and classification of tangent harmonic maps at extreme black hole horizons.
The paper explores symmetries and conserved charges on pre-symplectic manifolds.
problem Analyzing conserved charges on solutions of Hamiltonian field theories.
method Using pre-symplectic structures and Gotay's coisotropic embedding theorem, the paper deals with gauge theories and examples like Electrodynamics and Klein-Gordon theory.
result Emergence of the energy-momentum tensor algebra of conserved currents.
Generalizes Hamiltonian structures to Dirac structures for new mechanics models.
problem Extending Hamiltonian structures to non-symplectic manifolds.
method Introduces Hamiltonian Lie algebroids and momentum sections over Dirac structures.
result Constructs new sigma models based on these structures.